When it is found that the line is erroneous, it is known that the distance run (by account) must be in error. Rules have been given for correcting the distance run in such cases, but as the line stretches unequally, in different parts, the only certain way is to measure the quantity run out in a measured time. See No. 260.

263. As the manner of heaving the log must be learned at sea, it is only necessary to remark, for reference, that the line is to be faked in the hand, not coiled; that the log-ship is to be thrown out well to leeward to clear the eddies near the wake, and in such a manner that it may enter the water perpendicularly, and not fall flat upon it; and that before a heavy sea the line should be paid out rapidly when the stern is rising, but when the stern is falling as this motion slacks the line, the reel should be retarded.

[2.] Massey's Log.

264. This instrument shews the distance actually gone by the ship through the water, by means of the revolutions of a fly towed astern, which are registered on a dial-plate. This log is highly approved in practice; and it is much to be desired that the patentee could manufacture, at a moderate price, an instrument which affords a method, at once so simple and so accurate, of measuring a ship's way, and which could not fail to come into extensive, if not general,

use.

[3] The Ground Log.

265. When the water is shoal, and the set of the tides or current much affected by the irregularity of the channel, or other causes; and when, at the same time, either the ship is altogether out of sight of land, or the shore presents no distinct objects by which to fix her position, recourse may be had to the ground log. This is a small lead, with a line divided like the log-line; the lead remaining fixed at the bottom, the line exhibits the effect of the combined motion of the ship through the water, and that of the water itself, or the current; and therefore the course (by compass) and distance made good are obtained at once.*

2. The Glasses.

266. The long glass runs out in 30 or in 28; the short glass runs out in half the time of the long one.

When the ship goes more than five knots, the short glass is used, and the number of knots shewn is doubled.

267. The sand-glasses should frequently be examined by a seconds watch, as in damp weather they are often retarded, and sometimes hang altogether. One end is stopped with a cork, which is taken out to dry the sand, or to change its quantity.

* In numerous passages up and down the river Plate, where the above circumstances toncur, made by H.M.S. Tyne, under the command of Captain Gordon Thomas Falcon, in 1818-19-20, we made constant use of this log.

G

268. When either the line or the glass is faulty, or when a line and glass not duly proportioned to each other are employed, the distance run is found as follows:

The number of feet in 1h is to the number of feet run out in an observed number of seconds, as 3600 (seconds in an hour) are to the observed number of seconds.

Ex. Suppose 190 feet of line are run out in 22*: required the rate.

The number of feet run out in 1h 190 :: 3600: 22; hence the number of feet 190 х 3600 31090 feet; which, divided by 6000 (as near enough), gives 5.2 miles.

22

On the occasion of a glass stopping, and in many other cases, it is a very useful acquirement to be able to count seconds for a small portion of time.

CHAPTER III.

THE SAILINGS.

1. PLANE SAILING, WITH TRAVERSE, CURRENT, AND Windward SAILINGS. II. PARALLEL SAILING, WITH MIDDLE LATITUDE, AND MERCATOR'S SAILINGS. III. GREAT CIRCLE SAILING.

269. In considering the place of a ship at sea, with reference to any other place which she has left, or to which she is bound, these five things are involved: the Course, Distance, Difference of Latitude, Departure, and Difference of Longitude.

270. In practice these two general questions occur.

1st. The course and distance from one place in given latitude and longitude to another are given, and it is required to find the latitude and longitude of the other place.

2d. The latitudes and longitudes of two places are given, and it is required to find the course and distance from one to the other.

The methods of solution, that is, the rules of calculation, by which the answers to such questions are obtained, are commonly termed SAILINGS.

I. PLANE SAILING.

271. In Plane Sailing, as the term implies, the path of the ship is supposed to be described on a plane surface.

If the ship sails 1 mile on a given course, she makes a certain D. lat. and Dep.; in sailing a second mile, on the same course, she

makes good the same D. lat. and Dep. as before. Thus the D. lat. and Dep. for 2 miles of Dist. are twice those for 1 mile; for 3 miles of Dist. they are three times those for 1 mile, and so on; that is, the total D. lat. and Dep. made good are proportional to the Dist. on the sphere as they would be on a plane. Plane Sailing, accordingly, treats of the relations of the Course, Dist., D. lat., and Dep., and applies to right-angled triangles generally.

But each mile of Dep. which the ship makes good corresponds to a Diff. of Long. which is different according to the latitude in which the ship moves (Note, p. 58), that is, there is no constant proportion between the Dep. and Diff. Long. in two different latitudes, and therefore a question in which Diff. Long. is concerned is not within the province of Plane Sailing, except the case in which the ship is on or near the equator, where Dep. and D. Long. are the same thing.

272. (I.) The proportions, No. 162, p. 46, as adapted to the figures, No. 200, p. 59 (or to the third figure of No. 163, where the course is the angle ABC), give the proportions or canons, as they are called, of Plane Sailing. We employ the following:

Dist. Dep. :: rad. (=1): sin. Co., whence, Dep.

Dist. D. Lat. ::

=

I : cos. Co.,

D. Lat. =

Dist. x sin. Co.
Dist. x cos. Co.

(1.)

(2.)

(2.) These equations* put into logarithms by the rules Nos. 64 and 65, p. 20, become

Which logarithmic equations contain the rules employed.

On ordinary occasions four places are enough.

Case I. Given the course and distance, to find the difference of latitude and departure.

Ex. 1. A ship sails N.W. by N. 103 miles from lat. 49° 30' N.; find the D. Lat. and Dep. and also the Lat. in.

273. By Inspection. Open Table 1 at 3 Points, and against the Dist. 103 stand D. Lat. 85.6 and Dep. 57 2.

Then 85-6 or 1° 25′-6 added to 49° 30′ gives Lat. in 50° 55'6 N.

=

* Quantities connected by the sign constitute an equation; thus 7 &c. are equations.

274. By Computation. (1.) For the D. Lat. To the log. cos. of the Course (Table 65 or 68) add the log. of the Dist. (Table 64); the sum (rejecting 10 from the index) is the log. of the D. Lat.

(2.) For the Dep. To the log. sine of the Course add the log. of the Dist.; the sum (rejecting 10) is the log. of the Dep.

Draw a line CN

275. By Construction. towards the north for the meridian. From the centre C, with the chord of 60° as radius, describe an arc on the west side of CN, and lay off the chord of three points, or 33° to a (No. 107). Through a draw Ca, this gives the angle N Ca equal to the Course, or three points; lay off from a scale of equal parts CA equal to the Dist. 103; draw A B perpendicular to CN, then C B will shew on the same scale the D. Lat. 85-6, and A B the Dep. 57-2.

Ex. 2. A ship sails S. 72° W. 216 miles from lat. 14° 11′ N.: required the D. Lat. and Dep., and also the Lat. in.

By Inspection. The Course 72° and Dist. 216 give D. LAT. 66.7 and DEP. 205.4. Then 66'7, or 1° 6'7, subtracted from 14° 11′ N. leaves Lat. in 13° 4′′3 N.

B

These two examples of construction are sufficient for all varieties of Case I. When the course is to the eastward, CA is drawn on the right side of the meridian C N or CS instead of the left side.

Case II. Given the course and difference of latitude, to find the distance and the departure.

Ex. 1. A ship sailing W.S.W. makes 47 miles D. Lat.: find the Dist. run and the Dep.

276. By Inspection. Enter Table 1 with the Course 6 points; look in the D. Lat. column for 47; the nearest to 47 is 47·1, against which stand the Dist. 123 and Dep. 113.6.

The Lat. of the ship is, from the nature of the case, already given.

277. By Computation. (1.) For the Dist. To the log. sec. of the Course add the log. of the D. Lat.; the sum (rejecting 10) is the log. of the Dist.

(2.) For the Dep. To the log. tan. of the Course add the log. of the D. Lat.; the sum (rejecting 10) is the log. of the Dep.

278. By Construction. Draw the meridian line CS; lay off the course, or angle SCA, 6 points (No. 107); from C lay off CB the D. Lat. 47; draw BA perpendicular to CS, then CA is the Dist. and A B the Dep.

This example will suffice for all varieties of Case II. When the course is to the northward, C N is drawn upwards instead of CS downwards; and when the course is to the eastward,

log. tan. 03828 log. 16721

log. 20549

(This is the Canon (3.) in No. 272.)

A

6 pts.

B

CA is to be drawn on the right side of the meridian instead of the left side.

Ex. 2. A ship sails N. 54° E. and makes 119 miles D. Lat. : required the Distance run and the Departure.

By Inspection. Course 54° in Table 2, and D. Lat. 119.3, give the DIST. 203 and DEP. 164.2, nearly enough in practice.

Case III. Given the difference of latitude and departure, to find the course and distance.

Ex. A ship makes 91 miles northing and 34°7 Dep. (easting): find her Course and Distance.

279. By Inspection. Look in Table 2 for 91 in the D. Lat. column, and 34.7 in the Dep. column; the nearest are 90.6 and 34.8, which give the Course 21° (N. 21° E. in this example) and Dist.

97 miles.

280. By Computation. (1.) For the Course. From the log. of the Dep. (adding 10 to its index if necessary) subtract the log. of the D. Lat.; the remainder is the log. tan. of the Course.

(2.) For the Dist. Find the Course; then to the log. sec. of the Course add the log. of the D. Lat.; the sum is the log. of the Dist.